Solving Groundwater Flow Problems with Time Series Analysis: You May Not Even Need Another Model

Time series analysis is a data-driven approach to analyze time series of heads measured in an observation well. Time series models are commonly much simpler and give much better fits than regular groundwater models. Time series analysis with response functions gives insight into why heads vary, while such insight is difficult to gain with black box models out of the artificial intelligence world. An important application is to quantify the contributions to the head variation of different stresses on the aquifer, such as rainfall and evaporation, pumping, and surface water levels. Time series analysis may be applied to answer many groundwater questions without the need for a regular groundwater model, such as what is the drawdown caused by a pumping station? Or, how long will it take before groundwater levels recover after a period of drought? Even when a regular groundwater model is needed to solve a groundwater problem, time series analysis can be of great value. It can be used to clean up the data, identify the major stresses on the aquifer, determine the most important processes that affect flow in the aquifer, and give an indication of the fit that can be expected. In addition, it can be used to determine calibration targets for steady-state models, and it can provide several alternative calibration methods for transient models. In summary, the overarching message of this paper is that it would be wise to do time series analysis for any application that uses measured groundwater heads.     

Analytic modeling of groundwater dynamics with an approximate impulse response function for areal recharge

An analytic approach is presented for the simulation of variations in the groundwater level due to temporal variations of recharge in surficial aquifers. Such variations, called groundwater dynamics, are computed through convolution of the response function due to an impulse of recharge with a measured time series of recharge. It is proposed to approximate the impulse response function with an exponential function of time which has two parameters that are functions of space only. These parameters are computed by setting the zeroth and first temporal moments of the approximate impulse response function equal to the corresponding moments of the true impulse response function. The zeroth and first moments are modeled with the analytic element method. The zeroth moment may be modeled with existing analytic elements, while new analytic elements are derived for the modeling of the first moment. Moment matching may be applied in the same fashion with other approximate impulse response functions. It is shown that the proposed approach gives accurate results for a circular island through comparison with an exact solution; both a step recharge function and a measured series of 10 years of recharge were used. The presented approach is specifically useful for modeling groundwater dynamics in aquifers with shallow groundwater tables as is demonstrated in a practical application. The analytic element method is a gridless method that allows for the precise placement of ditches and streams that regulate groundwater levels in such aquifers; heads may be computed analytically at any point and at any time. The presented approach may be extended to simulate the effect of other transient stresses (such as fluctuating surface water levels or pumping rates), and to simulate transient effects in multi-aquifer systems.     

Modeling time series of ground water head fluctuations subjected to multiple stresses

The methods behind the predefined impulse response function in continuous time (PIRFICT) time series model are extended to cover more complex situations where multiple stresses influence ground water head fluctuations simultaneously. In comparison to autoregressive moving average (ARMA) time series models, the PIRFICT model is optimized for use on hydrologic problems. The objective of the paper is twofold. First, an approach is presented for handling multiple stresses in the model. Each stress has a specific parametric impulse response function. Appropriate impulse response functions for other stresses than precipitation are derived from analytical solutions of elementary hydrogeological problems. Furthermore, different stresses do not need to be connected in parallel in the model, as is the standard procedure in ARMA models. Second, general procedures are presented for modeling and interpretation of the results. The multiple-input PIRFICT model is applied to two real cases. In the first one, it is shown that this model can effectively decompose series of ground water head fluctuations into partial series, each representing the influence of an individual stress. The second application handles multiple observation wells. It is shown that elementary physical knowledge and the spatial coherence in the results of multiple wells in an area may be used to interpret and check the plausibility of the results. The methods presented can be used regardless of the hydrogeological setting. They are implemented in a computer package named Menyanthes (www.menyanthes.nl).     

Three-dimensional geostatistical inversion of synthetic tomographic pumping and heat-tracer tests in a nested-cell setup

A main purpose of groundwater inverse modeling lies in estimating the hydraulic conductivity field of an aquifer. Traditionally, hydraulic head measurements, possibly obtained in tomographic setups, are used as data. Because the groundwater flow equation is diffusive, many pumping and observation wells would be necessary to obtain a high resolution of hydraulic conductivity, which is typically not possible. We suggest performing heat tracer tests using the same already installed pumping wells and thermometers in observation planes to amend the hydraulic head data set by the arrival times of the heat signals. For each tomographic combinations of wells, we recommend installing an outer pair of pumping wells, generating artificial ambient flow, and an inner well pair in which the tests are performed. We jointly invert heads and thermal arrival times in 3-D by the quasi-linear geostatistical approach using an efficiently parallelized code running on a mid-range cluster. In the present study, we evaluate the value of heat tracer versus head data in a synthetic test case, where the estimated fields can be compared to the synthetic truth. Because the sensitivity patterns of the thermal arrival times differ from those of head measurements, the resolved variance in the estimated field is 6 to 10 times higher in the joint inversion in comparison to inverting head data only. Also, in contrast to head measurements, reversing the flow field and repeating the heat-tracer test improves the estimate in terms of reducing the estimation variance of the estimate. Based on the synthetic test case, we recommend performing the tests in four principal directions, requiring in total eight pumping wells and four intersecting observation planes for heads and temperature in each direction.     

Effect of the inter-annual variability of rainfall statistics on stochastically generated rainfall time series: part 2. Impact on watershed response variables

This study analyzes how the stochastically generated rainfall time series accounting for the inter-annual variability of rainfall statistics can improve the prediction of watershed response variables such as peak flow and runoff depth. The modified Bartlett–Lewis rectangular pulse (MBLRP) rainfall generation model was improved such that it can account for the inter-annual variability of the observed rainfall statistics. Then, the synthetic rainfall time series was generated using the MBLRP model, which was used as input rainfall data for SCS hydrologic models to produce runoff depth and peak flow in a virtual watershed. These values were compared to the ones derived from the synthetic rainfall time series that is generated from the traditional MBLRP rainfall modeling. The result of the comparison indicates that the rainfall time series reflecting the inter-annual variability of rainfall statistics reduces the biasness residing in the predicted peak flow values derived from the synthetic rainfall time series generated using the traditional MBLRP approach by 26–47 %. In addition, it was observed that the overall variability of the peak flow and run off depth distribution was better represented when the inter-annual variability of rainfall statistics are considered.     

Rainfall stochastic disaggregation models: Calibration and validation of a multiplicative cascade model

The simulation of long time series of rainfall rates at short time steps remains an important issue for various applications in hydrology. Among the various types of simulation models, random multiplicative cascade models (RMC models) appear as an appealing solution which displays the advantages to be parameter parsimonious and linked to the multifractal theory. This paper deals with the calibration and validation of RMC models. More precisely, it discusses the limits of the scaling exponent function method often used to calibrate RMC models, and presents an hydrological validation of calibrated RMC models. A 8-year time series of 1-min rainfall rates is used for the calibration and the validation of the tested models. The paper is organized in three parts. In the first part, the scaling invariance properties of the studied rainfall series is shown using various methods (q-moments, PDMS, autocovariance structure) and a RMC model is calibrated on the basis of the rainfall data scaling exponent function. A detailed analysis of the obtained results reveals that the shape of the scaling exponent function, and hence the values of the calibrated parameters of the RMC model, are highly sensitive to sampling fluctuation and may also be biased. In the second part, the origin of the sensivity to sampling fluctuation and of the bias is studied in detail and a modified Jackknife estimator is tested to reduce the bias. Finally, two hydrological applications are proposed to validate two candidate RMC models: a canonical model based on a log-Poisson random generator, and a basic micro-canonical model based on a uniform random generator. It is tested in this third part if the models reproduce faithfully the statistical distribution of rainfall characteristics on which they have not been calibrated. The results obtained for two validation tests are relatively satisfactory but also show that the temporal structure of the measured rainfall time series at small time steps is not well reproduced by the two selected simple random cascade models.     

采用主成分分析方法研究降水对地下水位的影响

基岩裂隙地下水的井水位变化是一个受综合因素影响的观测量。 文中采用主成分分析方法, 分析了福建晋江青阳基岩裂隙水观测井2007年3~8月的地下水位观测资料, 并分别与降水当天值和降水累加值进行了对比研究, 讨论了两种计算结果。 根据计算结果分析认为, 基岩裂隙水观测井的地下水位对当天降水值的响应不显著, 主成分分析中的第一综合量主要为水位本身的得分, 而第二种情况得到的结果则分别反映了3~8月份逐月降水累加值对水位的影响, 说明降水对水位的影响具有累加效应, 并对5~8月份造成这种累加效应不明显的情况进行了探讨, 给出了逐月水位与降水累加值的逐月第一与第二综合量曲线。     

The relationship between runoff rate and lag time and the effects of surface treatments at the plot scale

Abstract

The manner in which both the seasonal and regional variations in storm duration, intensity and inter-storm period manifest in the runoff response of agricultural water supply catchments is investigated. High-resolution rainfall data were analysed for a network of 17 raingauges located across the semiarid (200–500 mm year^?1) agricultural districts of southwest Western Australia. Seasonal variations in mean storm duration, mean rainfall intensity and mean inter-storm period were modelled using simple periodic functions whose parameters were then also regressed with geographic and climatic indices to create spatial fields for each of these statistics. Based on these mean values, a continuous rainfall time series can be synthesized for any location within the region, with the rainfall depth within each storm being downscaled to 5-min time steps using a bounded random cascade model. Runoff from six different catchment surface treatments (“engineered” catchments) was simulated using a conceptual water-balance model, validated using rainfall—runoff data from an experimental field site. The expected yield of the various catchment types at any other location within the study region is then simulated using the above rainfall—runoff model and synthetic rainfall and potential evaporation time series under a range of climatic settings representative of regional climate variation. The resulting coupled model can be used to estimate the catchment area required to yield an acceptable volume of runoff for any location and dam capacity, at a specified reliability level, thus providing a tool for water resource managers to design engineered catchments for water supply. Although the model presented is specific for Western Australia's southwest region, the methodology itself is applicable to other locations.     

Investigating the time dynamics of monthly rainfall time series observed in northern Lebanon by means of the detrended fluctuation analysis and the Fisher-Shannon method

We investigate the time dynamics of monthly rainfall series intermittently recorded on seven climatic stations in northern Lebanon from 1939 to 2010 using the detrending fluctuation analysis (DFA) and the Fisher-Shannon (FS) method. The DFA is employed to study the scaling properties of the series, while the FS method to analyze their order/organization structure. The obtained results indicate that most all the stations show a significant persistent behavior, suggesting that the dynamics of the rainfall series is governed by positive feedback mechanisms. Furthermore, we found that the Fisher Information Measure (the Shannon entropy power) seems to decrease (increase) with the height of the rain gauge; this indicates that the rainfall series appear less organized and less regular for higher-located stations. Such findings could be useful for a better comprehension of the climatic regimes governing northern Lebanon.     

CMBEAR: Python-Based Recharge Estimator Using the Chloride Mass Balance Method in Australia

The chloride mass balance (CMB) method is widely used to estimate long-term rates of groundwater recharge. In regions where surface water runoff is negligible, recharge can be estimated using measurements of chloride concentrations of groundwater and precipitation, and an estimate of long-term average rainfall. This paper presents the Chloride Mass Balance Estimator of Australian Recharge (CMBEAR), a Jupyter (Python) Notebook that is set up to rapidly apply the CMB method using gridded maps of chloride deposition rates across the Australian continent. For an Australian context, the chloride deposition rate and rainfall maps have been provided. Thus, CMBEAR requires only a spreadsheet with the groundwater chloride concentration, the latitude and longitude of the sample location, and some simple user inputs. CMBEAR may be easily applied in other regions, providing that a gridded chloride deposition map is available. Recharge estimates from CMBEAR are compared against published applications of the CMB method. CMBEAR is also applied to a large dataset from the Northern Territory and is used to produce a gridded map of recharge for western Victoria. CMBEAR provides a reproducible and straightforward approach to apply the CMB method to estimate groundwater recharge.     

Geomorphic effects of a large flood on fluvial fans

The response of 12 fluvial fans near Sydney, Australia to a large storm between 2 and 4 February 1990 was determined by repeating previously surveyed longitudinal profiles and by undertaking detailed field observations of erosion and deposition. Peak rainfall intensities occurred on 3 and 4 February when between 173 and 193·8 mm were recorded. Return periods for 24 h duration peak rainfall ranged between 5·7 and 11·0 years on the annual maximum series at six stations within the study area and return periods for 48 h peak rainfall ranged between 13·5 and 29·4 years. Of the 12 fans, seven were trenched and five untrenched. The most significant geomorphic effects of the storm were recorded on the proximal region of the fans. However, fan response was highly variable, with one fan exhibiting no detectable change, three fans localized deposition, two fans spatially disjunct erosion and deposition, two fans channel avulsions, and seven fans fanhead trench reworking. Some fans exhibited more than one type of response. A four-stage, tentative cyclical model of fanhead development was constructed from the field data. Stage 1 refers to the episodic aggradation of the fanhead by localized deposition, spatially disjunct erosion and deposition and/or channel avulsions. Stage 2 represents the initiation of a fanhead trench when progressive aggradation locally exceeds a threshold slope leading to localized erosion. This erosion initially creates one or more discontinuous flow-aligned scour pools. Over time, the scour pools widen, deepen and extend both up- and downfan. Stage 3 refers to the coalescence of discontinuous scour pools into a continuous trench by the removal of intervening log and boulder steps. Stage 4 represents the backfilling phase of the trench once it has been overwidened and/or slope reduced. Aggradation then continues as for stage one.     

Efficient conceptual model for simulating the effect of aquifer heterogeneity on natural groundwater discharge to rivers

When linearity can be assumed (linear response of heads to stresses), stream–aquifer flow exchange can be simulated as the drainage of a number of independent linear reservoirs. This conceptual model, which can be mathematically deduced in a univocal way from an eigenvalue solution of the linear groundwater flow problem, facilitates the understanding of the physical phenomenon and the analysis of influencing factors. The number of reservoirs required to simulate stream depletion in some ideal homogeneous cases of stream–aquifer connection was analyzed in detail in a previous investigation using analytical eigenvalue solutions [16]. However, most aquifers are heterogeneous in nature and numerical solutions must be employed to analyze whether they could also be simulated using few reservoirs. This paper presents a stochastic analysis of the influence of heterogeneity on the simulation of natural groundwater discharges in aquifers connected to rivers, as a series of linear reservoirs. A Monte-Carlo approach was employed to perform this study. The results show that, on a monthly time scale, many cases (even heterogeneous aquifers) can be simulated using just a few reservoirs with sufficient accuracy and at minimum computational cost. Therefore, this modeling technique can be useful to efficiently simulate the integrated management of complex water resources systems at the basin scale (with many aquifers, reservoirs, demands, etc.) that need to simultaneously consider surface and groundwater flow and stream–aquifer interaction.     

A comparison of two stochastic inverse methods in a field-scale application

Inverse modeling is a useful tool in ground water flow modeling studies. The most frequent difficulties encountered when using this technique are the lack of conditioning information (e.g., heads and transmissivities), the uncertainty in available data, and the nonuniqueness of the solution. These problems can be addressed and quantified through a stochastic Monte Carlo approach. The aim of this work was to compare the applicability of two stochastic inverse modeling approaches in a field-scale application. The multi-scaling (MS) approach uses a downscaling parameterization procedure that is not based on geostatistics. The pilot point (PP) approach uses geostatistical random fields as initial transmissivity values and an experimental variogram to condition the calibration. The studied area (375 km2) is part of a regional aquifer, northwest of Montreal in the St. Lawrence lowlands (southern Québec). It is located in limestone, dolomite, and sandstone formations, and is mostly a fractured porous medium. The MS approach generated small errors on heads, but the calibrated transmissivity fields did not reproduce the variogram of observed transmissivities. The PP approach generated larger errors on heads but better reproduced the spatial structure of observed transmissivities. The PP approach was also less sensitive to uncertainty in head measurements. If reliable heads are available but no transmissivities are measured, the MS approach provides useful results. If reliable transmissivities with a well inferred spatial structure are available, then the PP approach is a better alternative. This approach however must be used with caution if measured transmissivities are not reliable.     

A cascade approach to continuous rainfall data generation at point locations

High-resolution temporal rainfall data sequences serve as inputs for a range of applications in planning, design and management of small (especially urban) water resources systems, including continuous flow simulation and evaluation of alternate policies for environmental impact assessment. However, such data are often not available, since their measurements are costly and time-consuming. One alternative to obtain high-resolution data is to try to derive them from available low-resolution information through a disaggregation procedure. This study evaluates a random cascade approach for generation of high-resolution rainfall data at a point location. The approach is based on the concept of scaling in rainfall, or, relating the properties associated with the rainfall process at one temporal scale to a finer-resolution scale. The procedure involves two steps: (1) identification of the presence of scaling behavior in the rainfall process; and (2) generation of synthetic data possessing same/similar scaling properties of the observed rainfall data. The scaling identification is made using a statistical moment scaling function, and the log–Poisson distribution is assumed to generate the synthetic rainfall data. The effectiveness of the approach is tested on the rainfall data observed at the Sydney Observatory Hill, Sydney, Australia. Rainfall data corresponding to four different successively doubled resolutions (daily, 12, 6, and 3 h) are studied, and disaggregation of data is attempted only between these successively doubled resolutions. The results indicate the presence of multi-scaling behavior in the rainfall data. The synthetic data generated using the log–Poisson distribution are found to exhibit scaling behaviors that match very well with that for the observed data. However, the results also indicate that fitting the scaling function alone does not necessarily mean reproducing the broader attributes that characterize the data. This observation clearly points out the extreme caution needed in the application of the existing methods for identification of scaling in rainfall, especially since such methods are also prevalent in studies of the emerging satellite observations and thus in the broader spectrum of hydrologic modeling.     

Identification and Explanation of a Change in the Groundwater Regime using Time Series Analysis

Time series analysis is applied to identify and analyze a transition in the groundwater regime in the aquifer below the sand ridge of Salland in the Netherlands, where groundwater regime refers to the range of head variations throughout the seasons. Standard time series analysis revealed a discrepancy between modeled and observed heads in several piezometers indicating a possible change in the groundwater regime. A new time series modeling approach is developed to simulate the transition from the initial regime to the altered regime. The transition is modeled as a weighted sum of two responses, one representing the initial state of the system, the other representing the altered state. The inferred timing and magnitude of the change provided strong evidence that the transition was the result of significant dredging works that increased the river bed conductance of the main river draining the aquifer. The plausibility of this explanation is corroborated by an analytical model. This case study and the developed approach to identify a change in the groundwater regime are meant to stimulate a more systematic application of time series analysis to detect and understand changes in groundwater systems which may easily go unnoticed in groundwater flow modeling.     

A class of hidden Markov models for regional average rainfall

Abstract

Basic hidden Markov models are very useful in stochastic environmental research but their ability to accommodate sufficient dependence between observations is somewhat limited. However, they can be modified in several ways to form a rich class of flexible models that are useful in many environmental applications. We consider a class of hidden Markov models that incorporate additional dependence among observations to model average regional rainfall time series. The focus of the study is on models that introduce additional dependence between the state level and the observation level of the process and also on models that incorporate dependence at observation level. Construction of the likelihood function of the models is described along with the usual second-order properties of the process. The maximum likelihood method is used to estimate the parameters of the models. Application of the proposed class of models is illustrated in an analysis of daily regional average rainfall time series from southeast and southwest England for the winter season during 1931 to 2010. Models incorporating additional dependence between the state level and the observation level of the process captured the distributional properties of the daily rainfall well, while the models that incorporate dependence at the observation level showed their ability to reproduce the autocorrelation structure. Changes in some of the regional rainfall properties during the time period are also studied.

Editor D. Koutsoyiannis     

Uncertainty based modeling of rainfall-runoff: Combined differential evolution adaptive Metropolis (DREAM) and K-means clustering

Simulation of rainfall-runoff process in urban areas is of great importance considering the consequences and damages of extreme runoff events and floods. The first issue in flood hazard analysis is rainfall simulation. Large scale climate signals have been proved to be effective in rainfall simulation and prediction. In this study, an integrated scheme is developed for rainfall-runoff modeling considering different sources of uncertainty. This scheme includes three main steps of rainfall forecasting, rainfall-runoff simulation and future runoff prediction. In the first step, data driven models are developed and used to forecast rainfall using large scale climate signals as rainfall predictors. Due to high effect of different sources of uncertainty on the output of hydrologic models, in the second step uncertainty associated with input data, model parameters and model structure is incorporated in rainfall-runoff modeling and simulation. Three rainfall-runoff simulation models are developed for consideration of model conceptual (structural) uncertainty in real time runoff forecasting. To analyze the uncertainty of the model structure, streamflows generated by alternative rainfall-runoff models are combined, through developing a weighting method based on K-means clustering. Model parameters and input uncertainty are investigated using an adaptive Markov Chain Monte Carlo method. Finally, calibrated rainfall-runoff models are driven using the forecasted rainfall to predict future runoff for the watershed. The proposed scheme is employed in the case study of the Bronx River watershed, New York City. Results of uncertainty analysis of rainfall-runoff modeling reveal that simultaneous estimation of model parameters and input uncertainty significantly changes the probability distribution of the model parameters. It is also observed that by combining the outputs of the hydrological models using the proposed clustering scheme, the accuracy of runoff simulation in the watershed is remarkably improved up to 50% in comparison to the simulations by the individual models. Results indicate that the developed methodology not only provides reliable tools for rainfall and runoff modeling, but also adequate time for incorporating required mitigation measures in dealing with potentially extreme runoff events and flood hazard. Results of this study can be used in identification of the main factors affecting flood hazard analysis.     

Multivariate analysis of soil moisture history for a hillslope

In this study, the spatial distribution of measured soil moisture was analyzed on the platform of multivariate modeling. Soil moisture time series for two seasons were selected and used for analysis to reveal similarities and differences in soil moisture responses for a few rainfall events. The development of a soil moisture transport process that considers the representative element volume and uncertainty of soil media provides the hydrological basis for time series modeling. The systematic procedure of Box–Jenkins with noise modeling was used to delineate the final models for all monitoring points. The physical basis of mass balance and the continuity in inflow contribution, as well as statistical criteria, were used in the model selection procedure. Heuristic approaches provide the spatial distribution of selected models along the transect of a hillside. Comparative analysis for two different depths and seasons provide an understanding of the variation in soil moisture transfer processes at the hillslope scale. Differences in soil moisture models for both depths and seasons are associated with eco-hydrological processes. The relationships between distributed topographic features and modeling results were explored to configure dominant hydrological processes for each season.